S O L U T I O N S
19
Distortionary Taxes and Subsidies
Solutions for Microeconomics: An Intuitive
Approach with Calculus (International Ed.)
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• Solutions to Within-Chapter Exercises are provided in the student Study Guide.
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Distortionary Taxes and Subsidies
19.1 In our discussion of economic versus statutory incidence, the text has focused primarily on the incidence of taxes. This exercise explores analogous issues related to the incidence of benefits from subsidies.
A: Consider a price subsidy for x in a partial equilibrium model of demand and supply in the market
for x.
(a) Explain why it does not matter whether the government gives the per-unit subsidy s to consumers or producers.
Answer: This is illustrated in panel (a) of Graph 19.1 where the pre-subsidy demand curve
D intersects the pre-subsidy supply curve S at price p ∗ and output level x ∗ . If the subsidy is
given to consumers, demand curves shift up by s to D ′ — causing the equilibrium to shift to
A with price p A and output level x s . If the subsidy is given to producers, supply shifts down
by s to S ′ — causing the equilibrium to shift to B with price p B and output level x s . The
output level in equilibrium A and B is therefore the same — and independent of whether
the subsidy was given to consumers or producers. The equilibrium price differs — but the
equilibrium impact on consumers and producers is identical. In equilibrium A, consumers
pay p A to producers but then receive s per unit back from the government — implying that
the suppliers’ price is p s = p A and consumer prices are p d = p A − s = p B . In equilibrium B ,
consumers pay p B to producers — and producers now receive a per unit subsidy s for each
good that is sold. Thus, the prices received by producers are p s = p B + s = p A and prices
paid by consumers are p d = p B . Whether the subsidy is given to producers or consumers,
it is therefore the case that p s = p A , p d = p B and output is x s .
Graph 19.1: The Economic Incidence of Subsidies
(b) Consider the case where the slopes of demand and supply curves are roughly equal in absolute
value at the no-subsidy equilibrium. What does this imply for the way in which the benefits
of the subsidy are divided between consumers and producers?
Answer: This is what we just illustrated in panel (a) of Graph 19.1 where the price received
by producers rises from p ∗ to p A and the price paid by consumers falls from p ∗ to p B .
When the slopes of the demand and supply curves are roughly equal in absolute value at
the initial equilibrium, the distance between p A and p ∗ will be roughly equal to the distance
between p ∗ and p B — i.e. consumers will benefit by about as much as producers will from
the subsidy.
(c) How does your answer change if the demand curve is steeper than the supply curve at the
no-subsidy equilibrium?
Answer: This is illustrated in panel (b) of Graph 19.1 where the consumer price p d drops
substantially more than the producer price p s rises from the initial p ∗ . Thus, when consumers are relatively less responsive to price changes, they will obtain the bulk of the benefit
from the subsidy.
Distortionary Taxes and Subsidies
684
(d) How does your answer change if the demand curve is shallower than the supply curve at the
no-subsidy equilibrium?
Answer: This is illustrated in panel (c) of Graph 19.1. Now the producer price p s rises more
than the consumer price p d falls from the initial p ∗ . Thus, when consumers are relatively
more responsive to price changes than producers, the bulk of the benefit from the subsidy
accrues to producers.
(e) Can you state your general conclusion — using the language of price elasticities — on how
much consumers will benefit relative to producers when price subsidies are introduced. How
is this similar to our conclusions on tax incidence?
Answer: The general conclusion is that the benefits of subsidies are shifted disproportionately to the side of the market that is less sensitive to price — i.e. the side of the market that
is more price inelastic. This is similar to what we concluded about the burden of a tax —
which is also shifted to the side of the market that is more price inelastic.
(f) Do any of your answers depend on whether the tastes for x are quasilinear?
Answer: No — our prediction of market outcomes in terms of prices and quantities are
based on uncompensated curves that include income and substitution effects — because
if we want to know what actually happens, we have to look at actual or uncompensated
curves. It is only when we try to assess changes in surplus — i.e. welfare changes — that we
need to use compensated (or marginal willingness to pay) curves. Had we drawn into our
graphs consumer surplus areas along the demand curves, we would therefore have had to
assume that the uncompensated demand curves in the graphs are also equal to marginal
willingness to pay curves — which would only be true if tastes are quasilinear in x.
B: In Section 19B.1, we derived the impact of a marginal per-unit tax on the price received by producers — i.e. d p s /d t .
(a) Repeat the analysis for the case of a per-unit subsidy and derive d p s /d s where s is the per-unit
subsidy.
Answer: Consider the general case where demand is given by x d (p), supply is given by x s (p)
and the no-tax equilibrium has price p ∗ and quantity x ∗ . Now suppose a small subsidy s (to
be received by consumers for each unit of x that is purchased) is introduced. This implies
that the price p d effectively paid by buyers is s lower than the price p s at which the good is
purchased from suppliers; i.e. p d = p s − s. Taking the differential of this, we get
d p d = d p s − d s;
(19.1)
i.e. the change in the consumer price p d is equal to the change in the producer price p s
minus the change in s. In the new equilibrium, demand has to equal supply, with each
evaluated at the relevant price; i.e.
x d (p d ) = x s (p s ).
(19.2)
Taking the differential of this, we can write
d xs
d xd
d pd =
d ps
d pd
d ps
(19.3)
and substituting equation (19.1) into equation (19.3), this becomes
d xd
d xs
(d p s − d s) =
d ps .
d pd
d ps
(19.4)
Rearranging terms in this equation, we can write it as
µ
¶
d xd
d xs
d xd
−
d s.
d ps =
d pd d p s
d pd
(19.5)
Before the subsidy is introduced, the equilibrium was at the intersection of supply and demand at p ∗ and x ∗ — a point on both the supply and demand curve. Multiplying equation
(19.5) by p ∗ /x ∗ , it becomes
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Distortionary Taxes and Subsidies
µ
¶
d xd p ∗
d xd p ∗ d x s p ∗
−
d s,
d ps =
d pd x ∗ d p s x ∗
d pd x ∗
(19.6)
which you should notice contains several price elasticity terms (evaluated at the no-tax
equilibrium). Rewriting the equation in terms of these price elasticities, it becomes
(εd − εs )d p s = εd d s
(19.7)
where εd is the price elasticity of demand and εs is the price elasticity of supply. Re-arranging
terms, we can also then write this as
εd
d ps
=
.
ds
εd − ε s
(19.8)
(b) What is d p d /d s?
Answer: Equation (19.1) implies that
d pd
ds
=
d ps
− 1.
ds
(19.9)
Substituting equation (19.8) into this equation, we then get
µ
¶
εd − ε s
εd
εd
d pd
εs
=
−1 =
−
=
ds
εd − ε s
εd − ε s
εd − ε s
εd − ε s
(19.10)
(c) What do your results in (a) and (b) tell you about the economic incidence of a per-unit subsidy
when the price elasticity of demand is zero? What about when the price elasticity of supply is
zero?
Answer: When εd = 0, we get
d pd
d ps
= −1 and
= 0;
ds
ds
(19.11)
i.e. d p d = −d s and d p s = 0. Thus, the entire benefit of the subsidy goes to consumers
whose price drops by the amount of the subsidy. When εs = 0, we get
d pd
d ps
= 0 and
= 1;
ds
ds
(19.12)
i.e. d p d = 0 and d p s = d s. Thus, the entire benefit of the subsidy goes to producers whose
price increases by the amount of the subsidy.
(d) What does your analysis suggest about the economic incidence of the subsidy when the price
elasticities of demand and supply are equal (in absolute value) at the no-subsidy equilibrium?
¯ ¯
Answer: In the case where ¯ε ¯ = εs , we get
d
d pd
εs
1
d ps
εd
1
=
= − and
=
= .
ds
εd − ε s
2
ds
εd − ε s
2
(19.13)
Thus, consumer prices fall by the same amount as producer prices increase — and the benefits of the subsidy are shared equally.
(e) More generally, can you show which side of the market gets the greater benefit when the absolute value of the price elasticity of demand is less than the price elasticity of supply?
¯ ¯
¯
¯
¯ ¯
Answer: When ¯ε ¯ < εs , ¯ε − εs ¯ > 2 ¯ε ¯. This implies
d
which implies
d
d
¯ ¯
¯ε ¯
1
¯ < ¯d ¯ =
¯
¯ε − ε s ¯ 2 ¯ε ¯ 2
d
d
¯ ¯
¯ε ¯
d
¯ ¯
¯ε ¯
εd
d ps
1
d
¯< .
=
=¯
¯
ds
εd − ε s
εd − ε s ¯ 2
(19.14)
(19.15)
Distortionary Taxes and Subsidies
686
Thus, producers will receive less than half the benefit of the subsidy when demand is more
price inelastic than supply — which is exactly our conclusion in part A(e). Put differently,
consumers receive a disproportionately larger share of the benefit of the subsidy when they
are more price inelastic than producers. (We could similarly derive d p d /d s < −0.5, which
says the same thing.)
687
Distortionary Taxes and Subsidies
19.2 In the chapter, we discussed the deadweight loss from taxes on consumption goods when tastes are
quasilinear in the taxed good, and we treated deadweight loss when tastes are not quasilinear for the case
of wage taxes. In this exercise, we will consider deadweight losses from taxation on consumption goods
when tastes are not quasilinear.
A: Suppose that x is a normal good for consumers.
(a) Draw the market demand and supply graph for x and illustrate the impact on prices (for
consumers and producers) and output levels when a per-unit tax t on x is introduced.
Answer: This is done in panel (a) of Graph 19.2 where the no-tax equilibrium has price
p ∗ and output level x ∗ . When a tax t is introduced, output falls to x t . The price paid by
consumers rises to p d and the price received by producers falls to p s .
Graph 19.2: Taxes and Deadweight Loss when x is normal
(b) Would your answer to (a) have been any different had we assumed that all consumers’ tastes
were quasilinear in x?
Answer: No, the answer would have been exactly the same. We predict the impact of policies on output and prices by using market demand and supply curves — not compensated
curves (which differ from uncompensated ones when goods are normal but not when goods
are quasilinear.)
(c) On a consumer diagram with x on the horizontal and “all other goods” (denominated in dollars) on the vertical axes, illustrate the impact of the tax on a consumer’s budget.
Distortionary Taxes and Subsidies
688
Answer: This is illustrated in the top graph of panel (b) of Graph 19.2 where the shallower
solid budget is the pre-tax budget with slope −p ∗ and the steeper solid budget is the posttax budget with slope −p d .
(d) In your graph from (c), illustrate the portion of deadweight loss that is due to this particular
consumer.
Answer: The after tax consumption level for this consumer occurs at A. This implies the
consumer pays a tax in the amount of T on all the goods she consumes. Had we taken
income away without distorting prices, we could have taken away an amount L (that would
have implied B as the optimal consumption level) without making the consumer any worse
off. The deadweight loss for this consumer is DW L = L − T .
(e) On a third graph, depict the demand curve for x for the consumer whose consumer diagram
you graphed in (d). Then illustrate on this graph the same deadweight loss that you first
illustrated in (d).
Answer: This is done in the lower graph in panel (b) of Graph 19.2 where the uncompensated demand curve d as well as the compensated demand curve, or marginal willingness
to pay curve MW T P , is derived from the top graph. The tax revenue T collected from this
consumer is then equal to area a while the lump sum amount that could have been taken
without making the consumer worse off than she is under the distortionary tax is L = (a +b).
This implies that the deadweight loss from this consumer is DW L = b.
(f) Now return to your graph from (a). Illustrate where deadweight loss lies in this graph. How
does it compare to the case where the original market demand curve arises from quasilinear
tastes rather than the tastes we are analyzing in this exercise?
Answer: The deadweight loss on the consumer side should now be measured on the aggregate compensated demand curve that passes through the post-tax price on the (uncompensated) demand curve. This is illustrated in panel (c) (below panel (a)) of Graph 19.2 by the
curve labeled D c . (The dashed demand and supply curves are those from panel (a).) We
would measure the consumer portion of deadweight loss as the area (b + c) if the uncompensated demand curve D were equal to the compensated demand curve (as it would be in
the quasilinear case). But, when x is a normal good, the deadweight loss on the consumer
side shrinks to just area (b). The deadweight loss on the producer side is unaffected by the
nature of consumer tastes — and continues to be equal to (d + e). Thus, overall deadweight
loss is now (b + d + e).
(g) True or False: We will overestimate the deadweight loss if we use market demand curves to
measure changes in consumer surplus from taxation of normal goods.
Answer: This is true. In panel (c) of Graph 19.2, we would overestimate the deadweight loss
by area c if we used the uncompensated market demand curve.
B: Suppose that consumers all have Cobb-Douglas tastes that can be represented by the utility function u(x, y ) = x α y (1−α) and each consumer has income I . Assume throughout that the price of y is
normalized to 1.
(a) Derive the uncompensated demand for x by a consumer.
Answer: Setting up the usual utility maximization problem and solving for x, the demand
function for x is
x(p, I ) =
αI
p
(19.16)
where p is the price of good x. (Recall that Cobb-Douglas tastes have the special property
that demands are functions of only that good’s price and not the prices of other goods —
thus the price of good y does not appear in the demand function for x).
(b) Suppose income is expressed in thousands of dollars and each consumer has income I = 2.5
(i.e. income of $2,500). There are 1000 consumers in the market. What is the market demand
function?
Answer: In that case, the market demand function would be
x d (p) = (1000)
2.5α 2500α
=
.
p
p
(19.17)
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Distortionary Taxes and Subsidies
(c) Suppose market supply is given by x s = βp. Derive the market equilibrium price and output
level.
Answer: Setting x s equal to x d and solving for p, we get equilibrium price p ∗ = 50(α/β)1/2 .
Plugging this back into either x d or x s , we get market output x ∗ = 50(αβ)1/2 .
(d) Suppose α = 0.4 and β = 10. Determine the equilibrium p d , p s and x t when t = 10. How do
these compare to what we calculated for the quasilinear tastes in Section 19B.2.1 (where we
assumed α = 1000 and β = 10) graphed in Graph 19.10?
Answer: When α = 0.4 and β = 10, the market demand and supply functions are
xd =
1000
and x s = 10p
p
(19.18)
which are identical to the market demand and supply functions in the quasilinear case of
the text graph(where we used the same market supply function and where we set α = 1000
for the market demand function x d = α/p that emerges from a representative agent with
quasilinear tastes).
Substituting p d = p s + 10 for the price in the demand function and p s for the price in the
supply function and then setting the two functions equal to one another, we can simplify to
get p 2s +10p s −100 = 0. Using the quadratic formula (and disregarding the negative answer),
we then get
p s ≈ 6.18 which implies p d ≈ 6.18 + 10 = 16.18.
(19.19)
Note that these are identical to what is shown in the text graph — which makes sense since
we have constructed market demand and supply curves as identical.
(e) What is the before-tax and after-tax quantity transacted?
Answer; Again, since the market demand and supply functions are the same as in the text
graph, we should get the same answers for x ∗ and x t — i.e.
x ∗ = 100 and x t ≈ 61.8.
(19.20)
You can of course check this by plugging the relevant prices into the demand or supply
functions.
(f) If you used the market demand and supply curves to estimate deadweight loss, what would it
be?
Answer: It would be exactly the same as what we calculated for the example in the text — i.e.
DW L = 172.19 (which appears in Table 19.1 and was also calculated in our answer to on of
the within-chapter exercises.) Of course the reason for this is again that the market demand
and supply curves are exactly as derived in the text — but the difference is that in the text
example they were derived from quasilinear tastes and thus the market demand curve was
also equal to the marginal willingness to pay curve of the representative consumer. This is
not the case here where tastes are Cobb-Douglas and thus not quasilinear.
(g) Calculate the real deadweight loss in this case — and explain why it is different than in Section 19B.2.1 where market demand and supply curves were the same as here.
Answer: The deadweight loss is
DW L = ∆C S + ∆P S − T R
(19.21)
where ∆C S is the change in consumer surplus resulting from the imposition of the tax, ∆P S
is the change in producer surplus resulting from the imposition of the tax and T R is the tax
revenue collected. Since the market demand and supply functions are the same here as they
are in the text, tax revenue will be the same — i.e.
T R = t x t = 10(61.8) = 618.
(19.22)
Similarly, the change in producer surplus (or profit) is the same as it was in the text examples
— i.e.
Distortionary Taxes and Subsidies
∆P S = 309.
690
(19.23)
The only difference emerges on the consumer side where the tastes in the text were quasilinear — which implied we could measure changes in consumer surplus along the uncompensated demand curve (because it is equal to the marginal willingness to pay curve in the
absence of income effects). But now — even though the uncompensated market demand
curve is identical to that in the text, the underlying tastes give rise to income effects. So we
need to re-calculate the change in consumer surplus. We can simply do this for one of the
1000 identical individuals — and then multiply by 1000. Thus, we need to calculate
¡
¢
∆C S = 1000 I − E(p ∗ ,u t ) .
(19.24)
We know that I = 2.5 and p ∗ = 10. But we haven’t yet calculated the expenditure function
or the post-tax level of utility u t . To calculate u t , we need to plug the post-tax consumption levels of x and y into the Cobb-Douglas utility function. Using the individual demand
x(p, I ) = αI /p, we get that x = 0.4(2.5)/16.18 ≈ 0.0618 at the after tax price p d = 16.18. Since
this costs 16.18(0.0618) = 1, it must be that y = 1.5 (since we assume the price of y is 1).
Thus,
u t = (0.0618)0.4 (1.5)0.6 ≈ 0.4189.
(19.25)
To find the expenditure function, we have to solve the problem
min px + y subject to u = x 0.4 y 0.6 .
x,y
(19.26)
Solving this in the usual way, we get compensated demand functions
x(p,u) =
µ
µ
¶
¶
3p 2/5
2 3/5
u and y (p,u) =
u.
3p
2
(19.27)
Plugging these into the objective function px + y , we get the expenditure function
E(p,u) = p
µ
µ
¶
¶
3p 2/5
2 3/5
5up 2/5
u+
u=
≈ 1.96p 2/5 u.
3p
2
33/5 22/5
(19.28)
Using these pieces, we can now revisit equation (19.24) and derive
¡
¢
∆C S = 1000 I − E(p ∗ ,u t ) = 1000(2.5 − E(10,0.4189))
´
³
= 1000 2.5 − 1.96(10)2/5 (0.4189) ≈ 438.
(19.29)
This implies a deadweight loss of DW L ≈ 438 + 309 − 618 = 129 as opposed to a deadweight
loss of 172 we would have estimated had we used the uncompensated demand curve. Put
differently, using the uncompensated demand curves would have led us to overestimate the
deadweight loss by about one third.
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Distortionary Taxes and Subsidies
19.3 In the text, we discussed deadweight losses that arise from wage taxes even when labor supply is
perfectly inelastic. We now consider wage subsidies.
A: Suppose that the current market wage is w ∗ and that labor supply for all workers is perfectly
inelastic. Then the government agrees to pay employers a per-hour wage subsidy of $s for every
worker hour they employ.
(a) Will employers get any benefit from this subsidy? Will employees?
Answer: Employers will get no benefit from the subsidy because the entire economic incidence will fall on employees whose labor supply is perfectly inelastic. Thus, wages paid by
employers will remain at w ∗ while wages received by employees will be (w ∗ + s).
(b) In a consumer diagram with leisure ℓ on the horizontal and consumption c on the vertical
axes, illustrate the impact of the subsidy on worker budget constraints.
Answer: This is illustrated in panel (a) of Graph 19.3.
Graph 19.3: Dead Weight Losses from Wage Subsidies with Inelastic Labor Supply
(c) Choose a bundle A that is optimal before the subsidy goes into effect. Locate the bundle that
is optimal after the subsidy.
Answer: This is also illustrated in panel (a) of the Graph — where C lies vertically above A
because the (uncompensated) labor response to the wage subsidy is perfectly inelastic.
(d) Illustrate the size of the subsidy payment S as a vertical distance in the graph.
Answer: This is again illustrated in panel (a) of Graph 19.3 where S is the vertical distance
between A and C .
(e) Illustrate how much P we could have paid the worker in a lump sum way (without distorting
wages) to make him just as well off as he is under the wage subsidy. Then locate the deadweight loss of the wage subsidy as a vertical distance in your graph.
Answer: This is also illustrated in panel (a) of Graph 19.3 where P is the difference between
the parallel budgets. The higher of these is the budget necessary to get the worker to his
post-subsidy utility level uC at the pre-subsidy wage w ∗ . The deadweight loss is then simply
the difference between S and P .
(f) On a separate graph, illustrate the inelastic labor supply curve as well as the before and aftersubsidy points on that curve. Then illustrate the appropriate compensated labor supply curve
on which to measure the deadweight loss. Explain where this deadweight loss lies in your
graph.
Distortionary Taxes and Subsidies
692
Answer: This is done in panel (b) of Graph 19.3. The subsidy causes workers to move from A
to C on their inelastic supply curve. The compensated labor supply curve that corresponds
to the after-subsidy utility level uC must then pass through C — and as long as there is at
least some substitutability between leisure and consumption, it must slope up (because it
only incorporates substitution effects). When measuring worker surplus on this compensated labor supply curve, we find surplus of (a + c) under the wage subsidy but only surplus
of (c) under the lump sum subsidy that eliminates the wage subsidy. Since the worker is
equally happy at B and C , the lump sum subsidy must therefore be equal to (a); i.e. P = a.
The actual wage subsidy paid, however, is s times l ∗ — which is equal to area (a + b); i.e.
S = a + b. Thus, the deadweight loss is DW L = b.
(g) True or False: As long as leisure and consumption are at least somewhat substitutable, compensated labor supply curves always slope up and wage subsidies that increase worker wages
create deadweight losses.
Answer: This is true. Substitution effects tell us that consumption of leisure decreases when
leisure becomes more expensive — which is equivalent to saying that labor supplied increases as w increases. Compensated labor supply curves only incorporate substitution
effects — and thus, so long as there is any substitutability between leisure and consumption, more labor will be supplied as w goes up. Put differently, compensated labor supply
curves must slope up — and it is that upward slope that gives rise to the deadweight loss
from wage subsidies.
B: Suppose that, as in our treatment of wage taxes, tastes over consumption c and leisure ℓ can be
represented by the utility function u(c,ℓ) = c α ℓ(1−α) and that all workers have leisure endowment
of L (and no other source of income). Suppose further that, again as in the text, the equilibrium
wage in the absence of distortions is w ∗ = 25.
(a) If the government offers a $11 per hour wage subsidy for employers, how does this affect the
wage costs for employers and the wages received by employees?
Answer: Since labor supply is perfectly inelastic for these tastes (as shown in the text), the
entire benefit of the subsidy accrues to workers. Thus, wages for workers increase to $36 per
hour while wages paid by employers remain unchanged at $25 per hour.
(b) Assume henceforth that α = 0.5. What is the utility level u s attained by workers under the
subsidy (as a function of leisure endowment L)?
Answer: From the utility maximization problem, we get that consumption demand is c =
36(0.5)L = 18L while leisure demand is 0.5L. (This is derived in the text — we simply plugged
in the after-subsidy wage of $36 and α = 0.5.) Plugging these back into the utility function,
we get
u s = (18L)0.5 (0.5L)0.5 = 3L.
(19.30)
(c) What’s the least (in terms of leisure endowment L) we would need to give each worker in a
lump sum way to get them to agree to give up the wage subsidy program?
Answer: In the text, we derived the compensated leisure demand and consumption demand
equations and, from these, the expenditure function
E(w,u) =
w (1−α) u
αα (1 − α)(1−α)
.
(19.31)
The expenditure necessary to get the worker to the utility level u s at the pre-subsidy wage
w = 25 is E(25,3L). Plugging in u = 3L, w = 25 and α = 0.5, we get E = 30L. Thus, the worker
would have to have 30L in order to be just as happy without the subsidy as he is with the
subsidy when the value of his leisure endowment is 25L. We would therefore have to give
the worker 5L in a lump sum way to make him as well off at a wage of $25 per hour as he is
under the subsidized wage of $36 per hour.
(d) What is the per worker deadweight loss (in terms of leisure endowment L) of the subsidy?
Answer: The deadweight loss is then the difference between what we actually have to pay in
a wage subsidy and what we could have paid in a lump sum way without making workers
worse off. We just concluded that a lump sum payment of 5L would be just as good for the
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Distortionary Taxes and Subsidies
worker as the wage subsidy. Under the wage subsidy, the government has to pay $11 per
hour worked — and workers always work 0.5L hours (taking the rest of their endowment
as leisure). Thus, the wage subsidy costs 11(0.5)L = 5.5L — implying a deadweight loss of
DW L = 0.5L per worker.
(e) Use the compensated labor supply curve to verify your answer.
Answer: In the text, we derived the compensated labor demand curve as
l sc (w,u) = L −
µ
¶
1−α α
u.
αw
(19.32)
Plugging in α = 0.5 and the after-subsidy utility level u − 3L, we get
l sc (w) = L −
µ
3L
w 0.5
¶
.
(19.33)
The equivalent lump sum payment is the area under this function between the before and
after-subsidy wage; i.e.
Lump Sum Payment =
µ
¶¸
Z36 ·
3L
L−
d w = 5L.
w 0.5
25
(19.34)
Subtracting this from the actual subsidy cost of 11(0.5L) = 5.5L, we get a deadweight loss of
DW L = 0.5L as before.
Distortionary Taxes and Subsidies
694
19.4 This exercise reviews some concepts from earlier chapters on consumer theory in preparation for
exercise 19.5.
A: Consider an individual saver who earns income now but does not expect to earn income in a
future period for which he must save.
(a) Draw a consumer diagram with current consumption c 1 on the horizontal axis and future
consumption c 2 on the vertical. Illustrate an intertemporal budget constraint assuming an
interest rate r — then draw an indifference curve that contains the optimal bundle A.
Answer: This is illustrated in panel (a) of Graph 19.4.
Graph 19.4: Compensated Savings Supply Curves
(b) Now suppose the interest rate increases to r ′. Illustrate the new budget constraint and indicate
where the new optimal bundle C will lie given that the individual does not change his savings
decision when interest rates change.
Answer: This is also illustrated in panel (a) of the graph where C lies directly above A since
savings — i.e. the amount not consumed now — remains unchanged.
(c) How much, in terms of future dollars, would this person be willing to pay to get the interest
rate to change from r to r ′ ? If he pays that amount, will he end up saving more or less?
Answer: The most this person is willing to pay for an increase in the interest rate is an
amount that will make him just as well off as he is under the lower interest rate — i.e. an
amount that puts him on indifference curve u A with a steeper budget that reflects the higher
interest rate r ′ . This is illustrated as the dashed budget in panel (a) of Graph 19.4 — and the
difference between the two parallel budgets is the most the individual would pay to get the
increase in the interest rate. If this happens, he will consume less now — i.e. save more.
(d) Suppose instead that the interest rate starts at r ′ and then falls to r . Illustrate how much I
would have to give this individual to compensate him for the drop in the interest rate. If this
is done, will he save more or less than he did at the high interest rate?
Answer: This is illustrated in panel (b) of Graph 19.4 where I would have to give the individual an amount that makes the low-interest rate budget tangent to the indifference curve u c .
The difference between the parallel budgets is the level of compensation necessary. If this
is done, the person will end up consuming more now than he did at the high interest rate —
i.e. he will save less.
(e) On a new graph, illustrate the individual’s inelastic savings supply curve. Then illustrate the
compensated savings supply curves that correspond to the utility levels the individual has at
the interest rates r and r ′ .
Answer: This is done in panel (c) of Graph 19.4. The compensated savings supply curves
illustrate changes in savings behavior that result solely from changes in interest rates leaving
695
Distortionary Taxes and Subsidies
utility unchanged. As we showed in the first two panels, if utility is held fixed and interest
rates change, savings move with the interest rate. Point B in panel (a), for instance, happens
at a higher interest rate — where current consumption falls from A and savings therefore
increases. Similarly, point D in panel (b) occurs at a lower interest rate than C — with more
current consumption and thus less savings.
(f) True or False: Compensated savings supply curves always slope up.
Answer: This is true. Compensated savings supply curves only incorporate substitution
effects — and substitution effects always indicate more savings as interest rates increase.
Thus, even though the uncompensated savings supply curves may be inelastic (or may
even slope down) because of counteracting wealth effects, the compensated savings supply curves must always slope up.
B: Suppose your tastes over current consumption c 1 and future consumption c 2 can be modeled
through the utility function u(c 1 ,c 2 ) = c 1α c 2(1−α) , your current income is I and you will earn no
income in the future. The real interest rate from this period to the future is r .
(a) Derive your demand functions c 1 (r, I ) and c 2 (r, I ) for current and for future consumption.
Answer: We have to solve utility maximization problem
max c 1α c 2(1−α) subject to c 2 = (1 + r )(I − c 1 ).
c 1 ,c 2
(19.35)
Solving this in the usual way, we get the demand functions for current and future consumption
c 1 = αI and c 2 = (1 − α)(1 + r )I .
(19.36)
(b) Define “savings” as the difference between current income and current consumption. Derive
your savings — or capital supply — function k s (r, I ). (Note: It turns out that this function is
not actually a function of r .)
Answer: The savings supply function is then simply
k s (r, I ) = (1 − α)I .
(19.37)
(c) Derive the indirect utility function V (r, I ) — i.e. the function that gives us your utility for any
combination of (r, I ).
Answer: We simply substitute the demand functions we just derived into the utility function
to get
V (r, I ) = (αI )α ((1 − α)(1 + r )I )(1−α) = αα (1 − α)(1−α) (1 + r )(1−α) I .
(19.38)
(d) Next, derive your compensated demand functions c 1c (r,u) and c 2c (r,u) for current and future
consumption.
Answer: For this, we have to solve the expenditure minimization problem which reverses
the objective function and constraint in the utility maximization problem. The objective
function of the expenditure minimization problem is therefore the budget constraint solved
for I — which is c 1 + c 2 /(1 + r ). The problem we have to solve is therefore
c2
subject to u = c 1α c 2(1−α) .
(1 + r )
Solving this in the usual way, we get
min c 1 +
c 1 ,c 2
(19.39)
µ
¶(1−α)
¶
(1 + r )(1 − α) α
α
(19.40)
u and c 2c (r,u) =
u
(1 + r )(1 − α)
α
(e) Define the expenditure function E(r,u) — i.e. the function that tells us the current income
necessary for you to reach utility level u at interest rate r .
Answer: We simply plug the compensated demands into the objective function of the expenditure minimization problem to get
c 1c (r,u) =
µ
E(r,u) =
u
αα (1 − α)(1−α) (1 + r )(1−α)
.
(19.41)
Distortionary Taxes and Subsidies
696
(f) Can you verify your answers by comparing V (r, I ) to E(r,u)?
Answer: Solving V (r, I ) for I (and replacing V (r, I ) with u) should give us the expenditure
function, and solving E(r,u) for u (and replacing E(r,u) with I ) should give us the indirect
utility function. This is indeed the case.
(g) Finally, suppose that we begin with an interest rate r and derive from it V (r , I ). Define the
compensated savings or compensated capital supply function as k sc (r,r ) = I − c 1c (r,V (r , I )).
Answer: We get
µ
¶(1−α)
α
αα (1 − α)(1−α) (1 + r )(1−α) I
(1 + r )(1 − α)
"
#
µ
¶
1 + r (1−α)
= 1−α
I.
1+r
k sc (r,r ) = I −
(19.42)
(h) What is the interest rate elasticity of savings? Without deriving it precisely, can you tell whether
the interest rate elasticity of compensated savings is positive or negative?
Answer: Since the savings function k s = (1 − α)I is not a function of r , savings is unaffected
by the interest rate — i.e. it is perfectly inelastic relative to the interest rate. The interest
rate elasticity of compensated savings will have the same sign as the derivative of k sc with
respect to r — which is
∂k sc
∂r
=
α(1 − α)(1 + r )(1−α) I
(1 + r )(2−α)
> 0.
(19.43)
Thus, the savings supply function is upward sloping — implying a positive interest rate elasticity.
697
Distortionary Taxes and Subsidies
19.5 (This exercise builds on exercise 19.4 which you should do before proceeding.) Through the income
tax code, governments typically tax most interest income — but, through a variety of retirement programs,
they often subsidize at least some types of interest income.
A: Suppose all capital is supplied by individuals that earn income now but don’t expect to earn income in some future period — and therefore save some of their current income. Suppose further
that these individuals do not change their current consumption (and thus the amount they put into
savings) as interest rates change.
(a) What is the economic incidence of a government subsidy of interest income? What is the
economic incidence of a tax on interest income?
Answer: The economic incidence of taxes and subsidies always falls more heavily on the
side of the market that behaves more inelastically. In this case, savers are perfectly inelastic
— which implies they will enjoy the full benefit of interest subsidies and pay the full cost of
interest taxes. (If you have trouble seeing this, draw a graph with the savings supply curve
almost perfectly inelastic — you should see that the incidence of taxes and subsidies then
falls almost entirely on savers.)
(b) In the text, we illustrated the deadweight loss from a subsidy on interest income when savings
behavior is unaffected by changes in the interest rate. Now consider a tax on interest income.
In a consumer diagram with current consumption c 1 on the horizontal and future consumption c 2 on the vertical axis, illustrate the deadweight loss from such a tax for a saver whose
(uncompensated) savings supply is perfectly inelastic.
Answer: This is done in panel (a) of Graph 19.5. The after tax optimal bundle (on the shallower budget) is A — giving utility u A . At that bundle, the saver pays the distance T in taxes.
But he would have been willing to pay up to L in order to keep the distortionary tax from being implemented — with the difference between L and T constituting the deadweight loss
from the tax on interest income.
Graph 19.5: Dead Weight Loss from Taxing Interest Income
(c) What does the size of the deadweight loss depend on? Under what special tastes does it disappear?
Answer: The size of the deadweight loss depends on the distance between A and B — which
in turn depends on the degree of substitutability between consumption now and in the future. The more complementary consumption is across time, the shorter this distance —
Distortionary Taxes and Subsidies
698
and the less the deadweight loss. If consumption across the two periods is perfectly complementary, B and A would be the same point — and the deadweight loss would disappear
because the substitution effect that gives rise to the deadweight loss would be gone. However, the uncompensated labor supply curve would then not be perfectly inelastic. Instead,
for the deadweight loss to disappear and the labor supply curve to be perfectly inelastic,
there would have to be a kink point at A that is sufficiently large to keep a substitution effect
from appearing while still allowing point C to lie directly above A in panel (a) of the graph.
(d) On a separate graph, illustrate the inelastic savings (or capital) supply curve. Then illustrate
the compensated savings supply curve that allows you to measure the deadweight loss from
the tax on interest income. Explain where in the graph this deadweight loss lies.
Answer: This is illustrated in panel (b) of Graph 19.5. The deadweight loss will be measured
on the compensated supply curve that corresponds to the after-tax utility level u A (and
therefore passes through A.) Saver surplus under the interest rate (r − t ) is just area b —
while saver surplus under the interest rate r is (a + b + c). But the saver is equally happy
at A and B because at B he had to pay a lump sum tax that pushes him to the indifference
curve u A at the pre-tax interest rate. This implies that the lump sum tax he is willing to pay
is (a + c). But the tax revenue that is collected under the distortionary tax is just area a. This
leaves us with deadweight loss c.
(e) What happens to the compensated savings supply curve as consumption becomes more complementary across time — and what happens to the deadweight loss as a result?
Answer: As consumption becomes more complementary across time, the distance between
A and B in panel (a) of the graph decreases — which also implies the distance between
C and B decreases in panel (b) of the graph Thus, the compensated savings supply curve
becomes more inelastic as consumption becomes more complementary across time — because the substitution effect becomes smaller. This causes the deadweight loss area c in the
graph to shrink. If consumption is perfectly complementary across time, C and B will lie on
top of one another in panel (b) of the graph — with the compensated savings supply curve
perfectly inelastic. In that case, the deadweight loss area disappears.
(f) Is the special case when there is no deadweight loss from taxing interest income compatible
with a perfectly inelastic uncompensated savings supply curve?
Answer: Yes and no. We would need consumption across time to be perfectly complementary — but that would imply that there is only a wealth effect and no substitution effect. And
this would imply that savings falls with an increase in the interest rate — i.e. C would lie to
the right of A in panel (a) of the graph. This further implies a downward sloping savings
supply curve, not a perfectly inelastic curve. But you could still have a kink at A that is not a
right angle and that still allows C to lie directly above A (as discussed in the answer to (c)).
B: Suppose everyone’s tastes and economic circumstances are the same as those described in part B
of exercise 19.4 — with α = 0.5 and I = 100,000.1
(a) Suppose further that there are 10,000,000 consumers like this — and they are the only source
of capital in the economy. How much capital is supplied regardless of the interest rate?
Answer: The (inelastic) supply of capital is given by
K s = 10,000,000k s = 10,000,000(1 − α)I = 10,000,000(0.5)(100, 000) = 500,000,000,000
(19.45)
— i.e. $500 trillion.
(b) Suppose next that demand for capital is given by K d = 25,000,000,000/r . What is the equilibrium real interest rate r ∗ in the absence of any price distortions?
Answer: Setting K d equal to the inelastic capital supply of $500,000,000,000,000 and solving
for r , we get r ∗ = 0.05.
1 Among other functions, you should have derived uncompensated and compensated savings function as
"
#
µ
¶
1 + r (1−α)
k s (r, I ) = (1 − α)I and k sc = 1 − α
(19.44)
I.
1+r
699
Distortionary Taxes and Subsidies
(c) Suppose that, for any dollar of interest earned, the government provides the person who earned
the interest a 50 cent subsidy. What will be the new (subsidy-inclusive) interest rate earned by
savers, and what will be the interest rate paid by borrowers? What if the government instead
taxed 50% of interest income?
Answer: Since the supply of savings is perfectly inelastic, savers will receive the full benefit
of the subsidy and bear the full burden of the tax. Thus, under the subsidy, the interest rate
earned by savers would be 1.5(r ∗ ) = 1.5∗0.05 = 0.075. Under the tax, the interest rate earned
by savers would fall to 0.5(r ∗ ) = 0.5(0.05) = 0.025. In both cases, borrowers would still pay
interest r ∗ = 0.05.
(d) Consider the subsidy introduced in (c). How much utility V will each saver attain under this
subsidy?
Answer: In exercise 19.4 we derived the indirect utility function as
V (r, I ) = αα (1 − α)(1−α) (1 + r )(1−α) I .
(19.46)
Substituting in α = 0.5, I = 100,000 and the post-subsidy interest rate for savers (r = 0.075),
we get V ≈ 51,841.1. (You can of course also simply use the demand functions for current
and future consumption to get c 1 = 50,000 and c 2 = 53,750 and plug these into the utility
function to get the same answer.)
(e) How much current income would each saver have to have in order to obtain the same utility
V at the pre-subsidy interest rate r ∗ ? In terms of future dollars, how much would it therefore
cost the government to make each saver as well off in a lump sum way as it does using the
interest rate subsidy?
Answer: In part (e) of exercise 19.4, we derived the expenditure function as
E(r,u) =
u
αα (1 − α)(1−α) (1 + r )(1−α)
.
(19.47)
We now simply need to plug α = 0.5, r = 0.5 and u = 51,841.1 to get E ≈ 101,183.47. Thus, in
current dollars, each saver’s income would have to increase by $1,183.47 — which is equal
to (1 + 0.05)1,183.47 ≈ $1,242.65.
(f) How much interest will the government have to pay to each saver (in the future) under the
subsidy? Use this and your previous answer to conclude the amount of deadweight loss per
saver in terms of future dollars. Given the number of savers in the economy, what is the overall
deadweight loss?
Answer: Since savers will always save $50,000, they will earn (1 + 0.075)50,000 = $3,750 in
interest income under the subsidy — which is $1,250 more than they would have earned
in the absence of the subsidy. Thus, the subsidy costs the government $1,250 per saver (in
terms of future dollars). Subtracting the lump sum payment that would have made savers
equally well off, we get a deadweight loss of
DW L = 1,250 − 1,242.65 = $7.35
(19.48)
per saver. Given that there are 10 million savers, the overall deadweight loss is therefore
about $73.5 million in terms next period.
(g) Derive the compensated savings function (as a function of r ) given the post-subsidy utility
level V .
Answer: In exercise 19.4 we derived the compensated savings function as
"
k sc (r,r ) = 1 − α
µ
#
¶
1 + r (1−α)
I
1+r
(19.49)
where r is the interest rate that determines the compensated utility level. If we want the
compensated savings function at the post-subsidy utility, we therefore set r to 0.075. Substituting in α = 0.5 and I = 100,000, we then get
k sc (r ) ≈ 100,000 −
51841.1
(1 + r )0.5
.
(19.50)
Distortionary Taxes and Subsidies
700
You can of course also derive this by substituting the compensated demand for current consumption evaluated at the post-subsidy utility level from I to get the same answer.
(h) Use your answer to (g) to derive the aggregate compensated capital supply function — and
then find the area that corresponds to the deadweight loss. Compare this to your answer in
part (f).
Answer: The aggregate compensated capital supply function is simply k sc multiplied by the
number of savers — i.e.
¶
µ
51841.1
K sc (r ) = 10,000,000 100,000 −
(1 + r )0.5
(19.51)
In panel (a) of Graph 19.6, the inverse of this — which we have called the compensated
capital supply curve — is depicted. This is analogous to Graph 19.7 in the text where we
concluded that area c in the graph is the deadweight loss. Panel (b) of the graph inverts this
back to plot the compensated capital supply function K sc (r ) — with areas from panel (a)
again labeled. We can now see that the deadweight loss area c is simply the rectangle (a +c)
minus the integral between r = 0.05 and r = 0.075. The rectangle (a + c) is simply the total
amount of capital K supplied under the subsidy multiplied by 0.025 — where K is simply
50,000 times the 10 million savers. Thus, (a + c)=500,000,000,000(0.025)=12,500,000,000.
Graph 19.6: Compensated Capital Supply Curve and Function
The deadweight loss area c is therefore
DW L = 12,500,000,000 − 10, 000, 000
= 12,500,000,000 − 10, 000, 000
= 73,531,136 ≈ $73.5 million.
Z 0.075 µ
0.05
h³
100,000 −
51841.1
(1 + r )0.5
¶
dr
´
i
100,000r − 2(51841.1)(1 + r )0.5 |0.075
0.05
(19.52)
This is precisely what we calculated in (f).
(i) Repeat parts (d) through (h) for the case of the tax on interest income described in part (c).
Answer: To get the after-tax utility level, we would substitute r = 0.025 rather than r = 0.075
into our expression for V to get V ≈ 50,621.14. To get the same utility level under the pretax interest rate r ∗ = 0.05, we would not substitute r = 0.025 (rather than r = 0.075) into
our expression for the expenditure function — as well as our new value for V . This gives us
701
Distortionary Taxes and Subsidies
E ≈ $98,802.35 — i.e. each saver would be willing to give up 100,000−98,802.35 = $1,197.65
in current dollars or 1,197.65(1 + 0.05) ≈ 1,257.53 in future dollars. The government would
actually receive only $1,250 in tax revenue (in terms of future dollars) — implying that individuals would be willing to give up $7.53 to avoid the tax. For 10 million individuals, that
amounts to a deadweight loss of approximately $75.3 million. To calculate the same deadweight loss using the compensated capital supply function, we can again use k sc (r,r ) defined in part (g) — but now r = 0.025 giving us the equation
k sc (r ) = 100,000 −
50621.14
(1 + r )0.5
.
(19.53)
Integrating this between r = 0.025 and r = 0.05, we get the lump sum payment an individual
would be willing to make to not incur the tax — which is again $1,257.53. Or, aggregating
the function across 10,000,000 individuals, integrating in the same way and then subtracting
the total tax payments received, we get approximately $75.3 million in deadweight loss.
(j) You have calculated deadweight losses for interest rates that are reasonable for 1-year time
horizons. If we consider distortions in people’s decisions over longer time horizons (such as
when they plan for retirement), a more reasonable time frame might be 25 years. With annual
market interest rates of 0.05 in the absence of distortions, can you use your compensated savings function (given in the footnote to the problem) to estimate again what the deadweight
losses from a subsidy that raises the effective rate of return by 50% and from a tax that lowers
it by 50% would be?
Answer: If the undistorted interest rate is 0.05 annually, then a dollar invested now will result
in (1 + 0.05)2 5 ≈ $3.39 — giving us an effective interest rate of 2.39 (or 239 percent) over the
25 years. If a subsidy raised that by 50%, it would raise it to approximately 3.585. Using our
compensated savings function, we would then get
#
¶
1 + r (1−α)
I
1+r
"
µ
¶ #
1 + 3.585 0.5
= 100,000 − 50,000
1+r
"
k sc (r ) = 1 − α
µ
= 100,000 −
107,063
(1 + r )0.5
.
(19.54)
Using the same steps as in (h), we then get the deadweight loss
DW L = 50,000(3.585 − 2.39)(10, 000, 000) − 10, 000, 000
¶
Z 3.585 µ
107,063
100,000 −
0.5
(1 + r )
2.39
(19.55)
= 45,019,422,909 ≈ $45 billion
or about $4,500 per individual. You can also verify this by using the expenditure function as
we did in the earlier parts of this exercise. For a tax that cuts the effective return from 2.39
to 1.195, on the other hand, we would use the compensated savings function
#
¶
1 + r (1−α)
I
1+r
"
µ
¶ #
1 + 1.195 0.5
= 100,000 − 50,000
1+r
"
k sc (r ) = 1 − α
µ
= 100,000 −
74,077.66
(1 + r )0.5
.
(19.56)
Subtracting the actual tax revenue from the appropriate integral (that gives us the lump sum
payment individuals would be willing to make to avoid the tax), we get
Distortionary Taxes and Subsidies
702
¶¸
·
Z 2.39 µ
74,077.66
− 1.195(50,000)(10, 000, 000)
DW L = 10,000,000
100,000 −
0.5
(1 + r )
1.195
= 64,671,207,082 ≈ $62.7 billion.
(19.57)
703
Distortionary Taxes and Subsidies
19.6 Business and Policy Application: City Wage Taxes: In the U.S., very few cities tax income derived
from wages while the national government imposes considerable taxes on wages (through both payroll
and income taxes) — and then passes some of those revenues back to city governments.
A: In this exercise, we will consider the reason for this difference in local and national tax policy —
and why city governments might in fact be “employing” the national government to levy wage taxes
and then have the national government return them to cities.
(a) Consider first a national labor market. While workers and firms can move across national
boundaries to escape domestic taxes, suppose that this is prohibitively costly for the labor
market that we are analyzing. Illustrate demand and supply curves for domestic labor (assuming that supply is upward sloping). Indicate the no-tax equilibrium wage and employment level and then show the impact of a wage tax.
Answer: This is illustrated in panel (a) of Graph 19.7 where the demand and supply curves
determine the employment level l ∗ in the absence of a wage tax. When the wage tax is
introduced, the employment level falls to l t , the wage paid by firms increases to w f and the
wage received by workers falls to w w .
Graph 19.7: Wage Taxes Nationally and Locally
(b) Next, consider a city government that faces a revenue shortfall and considers introducing a
wage tax. Why might you think that labor demand and supply are more elastic from the city’s
perspective than they are from a national government perspective?
Answer: We would expect labor demand and supply to be more elastic because workers
and firms can move between cities in response to adverse conditions. Thus, we would expect firms to respond more elastically to changes in costs (as when wage taxes are passed
to firms), and we would similarly expect workers to respond more elastically in terms of
their labor supply to wages (as they can work in nearby cities that do not impose taxes, for
instance).
(c) Given your answer to (b), draw two Laffer curves — one for tax revenue raised in a city when
the tax is imposed nationally and one for tax revenues raised in the same city when it is imposing the tax on its own. Explain where the peaks of the two Laffer curves are relative to one
another.
Answer: This is illustrated in panel (c) with the aid of panel (b) of Graph 19.7. In panel (b),
we illustrate a solid set of demand and supply curves (similar to those in panel (a)) and a
shallower set of dashed demand and supply curves (D ′ and S ′ ) that intersect at the same
no-tax equilibrium but are more elastic. We then show the tax revenue box for a wage tax
t on the shallower demand curves — and the same considerably larger tax revenue box
Distortionary Taxes and Subsidies
704
extending to the second vertical bold line. The second box is so much larger than the first
because the impact of the same sized tax causes employment to drop to l ′′ when the curves
are more elastic but only to l ′ when the curves are more inelastic. This illustrates that tax
revenue diminishes much more quickly when demand and supply curves are more elastic
— as they are from the city perspective as opposed to the national perspective. And this in
turn implies a relationship of Laffer curves for city revenues when the tax is national versus
when the tax is local must be like that drawn in panel (c).
(d) How do your answers to (b) and (c) most likely contain the answer to why cities do not typically use wage taxes to raise revenues?
Answer: Cities simply cannot raise much revenue from wage taxes without driving workers
and firms to nearby cities. This accounts for the greater elasticity of labor demand and
supply when the city imposes wage taxes by itself — and for the less favorable Laffer curve.
(e) Suppose you are a mayor of a city and would like to impose a wage tax but understand the
problem so far. How might it make sense for you to ask the federal government to increase the
wage tax nationwide — and then to give cities the additional revenue collected in each city?
Answer: By forcing all cities to impose wage taxes by having the national government do it
uniformly, the elasticity of labor supply and demand that comes from the ability of firms
and workers to move in response to unfavorable local tax conditions is eliminated. Thus,
the more favorable Laffer Curve is restored.
(f) Of those cities that do have wage taxes, most are relatively large. Why do you think it is exceedingly rare for small cities to impose local wage taxes?
Answer: The larger the city, the harder it might be for many firms to move to nearby cities or
for workers to work in nearby cities. The smaller the city, the easier it is for firms to respond
to local conditions by moving. Thus, one would expect the demand and supply for labor
to become increasingly shallow as the city gets smaller — making it nearly impossible to
impose a local wage tax of much significance.
(g) Does any of this analysis depend on whether there are wealth (or income) effects in the labor
market?
Answer: No — the entire analysis is about predicting employment levels, wages and tax
revenues. For all of these, we need to work with (uncompensated) market demand and
supply curves. The only need to think about wealth effects would arise if we were to label
workers’ surplus in our graphs — in which case we would need to derive the compensated
labor supply curve.
B: Suppose that labor demand and supply are linear — with l d = (A − w)/α and l s = (w − B )/β.
(a) For a given per-unit wage tax t , calculate the employment level and tax revenue.
Answer: These demand and supply curves are identical to those used within the text in the
goods market. Letting w f = w w +t (where w f is the wage paid by firms and w w is the wage
received by workers), we can write the demand function as l d = (A−w w −t )/α and the labor
supply function as l s = (w w − B )/β. Setting these equal to each other and solving for w w ,
we get
ww =
βA + αB − βt
.
α+β
(19.58)
Substituting this back into either the labor supply or demand equation and simplifying, we
then get
lt =
A −B −t
α+β
(19.59)
and tax revenue
T Rt = t l t =
(A − B )t − t 2
.
α+β
(19.60)
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Distortionary Taxes and Subsidies
(b) Consider two scenarios — scenario 1 in which (A −B ) is large and scenario 2 in which (A −B )
is small. What has to be true about (α + β) in scenario 1 relative to scenario 2 if the no-tax
equilibrium employment level is the same in both cases.
Answer: Setting t = 0 in equation (19.59), we get the no-tax employment level
x∗ =
A −B
.
α+β
(19.61)
Thus, if x ∗ remains the same in the two scenarios, it must be that (α +β) falls as (A −B ) gets
smaller. Put differently, if the vertical intercepts of the demand and supply curves get closer
to one another (as in panel (b) of Graph 19.7), it must be that the demand and supply curves
become shallower in order for them to intersect at the same employment level.
(c) Suppose one scenario is relevant for predicting tax revenue from your city when it is collected
nationwide and the other is relevant for predicting tax revenue when the wage tax is collected
just in your city. Which scenario belongs to which tax analysis?
Answer: The scenario with smaller (A − B ) (and smaller (α + β) if employment levels are
unchanged in the two scenarios when t = 0) is the appropriate one to consider when the
city imposes the wage tax by itself — because this scenario corresponds to more elastic labor
demand and supply.
(d) Find the tax rate t at which government revenue is maximized.
Answer: We have already derived in equation (19.60) the relationship between the tax rate
t and tax revenue T R (i.e. the Laffer curve). To determine the rate at which tax revenue is
maximized, we have to take the derivative of T R with respect to t , set it to zero and solve for
t . This gives us
t=
A −B
.
2
(19.62)
(e) Demonstrate that the scenario appropriate for the tax analysis when only your city imposes
the wage tax leads to a Laffer Curve that peaks earlier.
Answer: The peak of the Laffer Curve occurs at t . We concluded above that the scenario
appropriate for the analysis where only your city imposes the wage tax is the one that corresponds to the smaller (A −B ) — which implies immediately that t is smaller in this scenario
than in the one applicable for a nationally imposed wage tax.
(f) As cities get small, what happens to (A − B ) in the limit? What happens to the peak of the
Laffer Curve for a local city tax in the limit?
Answer: As cities get small, the labor demand and supply curves should flatten out — with
A converging to B and (A − B ) converging to zero. As a result, the peak t of the Laffer curve
converges to zero as cities get small.
Distortionary Taxes and Subsidies
706
19.7 Business and Policy Application: Land Use Policies: In most Western democracies, it is settled law
that governments cannot simply confiscate land for public purposes. Such confiscation is labeled a “taking” — and, even when the government has compelling reasons to “take” someone’s property for public use,
it must compensate the landowner. But, while it is clear that a “taking” has occurred when the government
confiscates private land without compensation, constitutional lawyers disagree on how close the government has to come to literally confiscating private land before the action constitutes an unconstitutional
“taking”.
A: Any restriction that alters the way land would otherwise be used reduces the annual rental value
of that land and, from the owner’s perspective, can therefore be treated as a tax on rental value.
(a) Explain why the above statement is correct.
Answer: There are two parts to the statement: First, restrictions on land use cause rental
values to decline, and second, that this is equivalent to a tax on land rents from the owner’s
perspective. Prior to any restrictions on the land, the user of the land (whether this is a
renter or the owner who can be though of as renting the land from himself) employs it in
the optimal way. If the restrictions placed on the land still permit the land to be used in this
way, the rental value of the land is unaffected. For instance, if the optimal use of the land
is to have an office building on it and a restriction is passed that prohibits users to plant
corn, nothing has really changed. But if the restriction impacts the use of the land, then by
definition we are no longer using it the way we would have in the absence of the restriction
— i.e. we are no longer employing it in the optimal way from the individual’s perspective.
Thus, the user gets less out of the land which lessens his demand for the land — i.e. the
rental value declines. From the owner’s perspective, it does not really matter what causes
the rental value to decline — whatever it is, this reduces the value of the land. Since taxing
land rents lowers land value, we can then set the tax just "right" so as to achieve a reduction
in land value — including the reduction that occurs as a result of a land use regulation.
(b) Suppose a land use regulation is equivalent (from the owner’s perspective) to a tax of t % on
land rents to be statutorily paid by landowners (where 0< t < 1). How does it affect the market
value of the land?
Answer: Under such a tax, the owners would still collect the same amount of land rent as
before but would then have to give up t % of it. This lowers the land rents that owners can
keep to (1 − t )% of what they kept originally — implying that land value (which is just the
present discounted value of all future land rents) falls by t %.
(c) I am about to buy an acre of land from you in order to build on it. Right before we agree on a
price, the government imposes a new zoning regulation that limits what I can do on the land.
Who is definitively made worse off by this?
Answer: You are definitively made worse off — because your land value drops immediately
by the net present value of the decrease in land rents caused by the regulation.
(d) Suppose you own 1000 acres of land that is currently zoned for residential development. Then
suppose the government determines that your land is home to a rare species of salamander —
and that it is in the public interest for no economic activity to take place on this land in order
to protect this endangered species. From your perspective, what approximate tax rate on land
rents that you collect is this regulation equivalent to? Do you think this is a “taking”?
Answer: Since you are effectively prohibited from using (or renting) the land, this is equivalent to you paying a 100% tax on land rents. Were the government to actually impose such a
tax, it would have essentially confiscated your land because all rents from it would go to the
government instead of to you. But from your perspective, the regulation to protect the salamanders is no different — you again lose all the value from your land. The only difference
is that the government has actually not taken possession of the value of the land the way it
would under the 100% land rent tax. But from the owner’s perspective, it sure looks like a
taking.
(e) Suppose that, instead of prohibiting all economic activity on your 1000 acres, the government
reduces your ability to build residential housing on it to a single house. How does your answer
change? What if it restricts housing development to 500 acres? Do you think this would be a
“taking”?
Answer: If you are now only allowed to build a single house, some small fraction of the original value of the land is retained — and the regulation therefore is not a complete “taking”
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Distortionary Taxes and Subsidies
from your perspective. But it’s pretty close to a complete taking. If the government restricts
housing development to only 500 of the 1000 acres, this is approximately equivalent to the
government taxing your land rents at 50% — or confiscating half your land. Courts would
almost certainly not call this a “taking” — but from the owner’s perspective, it’s just like the
government just took half the land.
B: Suppose that people gain utility from housing services h and other consumption x, with tastes
described by the utility function u(x,h) = ln x + ln h. Consumption is denominated in dollars (with
price therefore normalized to 1). Housing services, on the other hand, are derived from the production process h = k 0.5 L α where k stands for units of capital and L for acres of land. Suppose 0 < α < 1.
Let the rental rate of capital be denoted by r , and assume each person has income of 1000.
(a) Write down the utility maximization problem and solve for the demand function for land
assuming a rental rate R for land.
Answer: Substituting the housing production function into the utility function, we can write
the utility function as
³
´
u(x,k,L) = ln x + ln k 0.5 L α = ln x + 0.5ln k + α ln L.
(19.63)
max ln x + 0.5ln k + α ln L subject to 1000 = x + r k + RL.
(19.64)
The utility maximization problem can then be written as
x,k,L
We can then write the Lagrangian function as
L = ln x + 0.5ln k + α ln L + λ (1000 − x − r k − RL) .
(19.65)
The first order conditions are then
0.5
α
1
= λ;
= λr and
= λR.
x
k
L
(19.66)
We can then use the first and third of these to write x in terms of L(i.e. x = RL/α) and we can
use the second and third of these to write k in terms of L (i.e. k = 0.5RL/(αr )). Substituting
these into the budget constraint, we can then solve for the demand function for land
L=
1000α
.
(1.5 + α)R
(19.67)
(b) Suppose your city consists of 100,000 individuals like this — and there are 25,000 acres of land
available. What is the equilibrium rental rate per acre of land (as a function of α)?
Answer: Setting demand equal to supply, we get the equation
100,000
µ
¶
1000α
= 25,000
(1.5 + α)R
(19.68)
that solves for
R∗ =
4000α
.
(1.5 + α)
(19.69)
(c) Using your answers above, derive the amount of land each person will consume.
Answer: Substituting equation (19.69) into (19.67), we get
L=
1
1000α(1.5 + α)
1000α
´=
³
=
4000α
(1.5
+
α)(4000α)
4
(1.5 + α) (1.5+α)
— i.e. everyone consumes a quarter of an acre in equilibrium.
(19.70)
Distortionary Taxes and Subsidies
708
(d) Suppose the government imposes zoning regulations that reduce the coefficient α in the production function from 0.5 to 0.25. What happens to the equilibrium rental value of land?
Answer: This implies that the equilibrium rental rate for an acre of land falls from
to
4000(0.5)
= 1000
(1.5 + 0.5)
(19.71)
4000(0.25)
≈ 571.43.
(1.5 + 0.25)
(19.72)
(e) Suppose that what you have calculated so far is the monthly rental value of land. What happens to the total value of an acre of land as a result of these zoning regulations assuming that
people use a monthly interest rate of 0.5% to discount the future?
Answer: This implies that land values fall from 1000/0.005 = $200,000 to 571.43/0.005 ≈
$114,286.
(f) Suppose that, instead of lowering α from 0.5 to 0.25 through regulation, the government imposes a tax t on the market rental value of land and statutorily requires renters to pay. Thus,
if the market land rental rate is R per acre, those using the land must pay t R on top of the rent
R for every acre they use. Set up the renters’ utility maximization problem, derive the demand
for land and aggregate it over all 100,000 individuals. Then derive the equilibrium land rent
per acre as a function of t (assuming α = 0.5).
Answer: The only thing that changes in the utility maximization problem is the rental price
in the budget constraint — which must now include the tax. Thus, the utility maximization
problem becomes
max ln x + 0.5ln k + 0.5ln L subject to 1000 = x + r k + (1 + t )RL.
x,k,L
(19.73)
The first order conditions are then
1
0.5
0.5
= λ;
= λr and
= λ(1 + t )R.
x
k
L
(19.74)
Solving these as before, we get the demand function for land
L=
250
.
(1 + t )R
(19.75)
Aggregating across 100,000 consumers and setting equal to supply, we get the equation
100,000
µ
¶
250
= 25,000.
(1 + t )R
(19.76)
Solving for R, we get the equilibrium rental rate
R∗ =
1000
.
(1 + t )
(19.77)
(g) Does the amount of land consumed by each household change?
Answer: Plugging the equilibrium land rental rate into equation (19.75), we get L = 1/4 as
before. So — no, the amount of land consumed by each household does not change —
because the tax inclusive rental rate for land remains unchanged. (Before, the land rental
rate was 1000 prior to the imposition of zoning; the tax inclusive rental rate now is (1+t )R ∗ =
1000.)
(h) Suppose you own land that you rent out. What level of t makes you indifferent between the
zoning regulation that drove α from 0.5 to 0.25 and the land rent tax that does not change α?
Answer: We calculated before that the rental rate on an acre of land falls to 571.43 under the
zoning regulation. You would therefore be indifferent between this and a land rent tax if the
rental rate under the land rent tax also fell to 571.43; i.e. if
709
Distortionary Taxes and Subsidies
This solves to t = 0.75.
1000
= 571.43.
(1 + t )
(19.78)
(i) Suppose the government statutorily collected the land rent tax from the owner instead of from
the renter. What would the tax rate then have to be set at to make the land owner indifferent
between the zoning regulation and the tax?
Answer: In this case, the consumer does not statutorily face a tax — and so the utility maximization problem is the same as in (a) leading to the same equilibrium rental rate R ∗ = 1000
as calculated in (b). Landowners would therefore collect $1,000 per acre but would then
have to pay t (1000) to the government. In order for this to leave them with an after-tax rent
of $571.43 (as under the zoning regulation), the tax rate would have to be set at t ≈ 0.4286.
Distortionary Taxes and Subsidies
710
19.8 Business and Policy Application: Price Floors for Corn: Is it a Tax or a Subsidy?: In exercises 18.9 and
18.10, we investigated policies that imposed a price floor in the corn market.
A: We will now see whether some of the price regulation proposals we considered are equivalent to
taxes or subsidies. For simplicity, assume that tastes are quasilinear in corn.
(a) In exercise 18.9, we began by considering a price floor without any additional government
program. Illustrate the equilibrium impact of such a price floor on the price of corn paid by
consumers as well as the price of corn received by producers.
Answer: This is illustrated in panel (a) of Graph 19.8. Consumers pay the legally mandated
price floor p while suppliers compete for the limited number of consumers by exerting additional effort. In equilibrium, the marginal effort cost must be t — lowering the price that
producers receive (net of the effort cost) to p s .
Graph 19.8: Price Floors as Taxes and Subsidies
(b) If you were to design a tax or subsidy policy that has the same impact as the stand-alone price
floor, what would it be?
Answer: Imposing a per unit tax of t will result in precisely the same prices for consumers
and producers — and the same reduction in output.
(c) In exercise 18.10, we considered the combination of a price floor and a government purchasing program under which the government guaranteed it would purchase any surplus corn at
the price floor and then sell it at a price sufficiently low for all of it to be bought. Illustrate the
impact of this program — including the deadweight loss.
Answer: This is illustrated in panel (b) of Graph 19.8 where the government purchases the
difference between what is demanded and what is supplied at the price floor p. Producers
therefore have to expend no additional effort and will receive a price equal to the price floor;
i.e. p s = p. In order for the government to sell its purchases of corn to consumers, however,
it will have to lower the price by s to p c . Those consumers who purchase at the price floor
will get consumer surplus of area a and producers will get surplus of (b + c + d + e). The
government takes a loss of (d +e + f +g ) because it buys at p and sells at p c — but consumers
who buy from the government get surplus of (d + f ). We therefore get total surplus of (a+d +
f ) for consumers, (b+c+d +e) for producers and a negative (d +e+ f +g ) for the government.
Adding all these together, we get societal surplus of (a + b + c + d − g ) — with deadweight
loss of g .
(d) If you were to design a tax or subsidy policy with the aim of achieving the same outcome for
the marginal consumer and producer as the policy in (c), what would you propose?
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Distortionary Taxes and Subsidies
Answer: You would choose a subsidy in the amount of s — which would increase output
by exactly the same amount and would result in the price p s for producers and p c for consumers.
(e) Would your proposal result in the same level of consumer and producer surplus? Would it
result in the same deadweight loss?
Answer: Consumer surplus would now be (a +b +d + f ), producer surplus would be (b +c +
d +e) and the government cost of the subsidy would be (b+d +e+ f +g ). Summing consumer
and producer surplus and subtracting the cost of the subsidy gives us (a +b +c +d −g ) — the
same overall surplus as under the price floor with government purchasing. The deadweight
loss is now also the same — i.e. area (g ).
B: Suppose, as in exercises 18.9 and 18.10, that the domestic demand curve for bushels of corn is given
by p = 24 − 0.00000000225x while the domestic supply curve is given by p = 1 + 0.00000000025x.
(a) Suppose the government imposes a price ceiling of p = 3.5 (as in exercise 18.9). In the absence
of any other program, how much will consumers pay (per bushel) and how much will sellers
keep (per bushel) after accounting for the additional marginal costs incurred by producers to
compete for consumers?
Answer: In exercise 18.9, we derived the equilibrium price (without price distortions) as
p ∗ = 3.3 — which implies that a price floor of 3.5 is binding (in the sense that p ∗ is illegal).
Consumers will therefore pay a price p c = 3.5 per bushel — and will buy x that solves the
equation 3.5 = 24 − 0.00000000225x — i.e. the quantity of corn sold will be x = 9,111,111
bushels. Producers will expend effort until the marginal producer makes zero profit —
which occurs when the effective price producers keep is
p s = 1 + 0.00000000025(9, 111, 111,111) ≈ 3.278.
(19.79)
(b) If you wanted to replicate this same outcome using taxes or subsidies, what policy would you
propose?
Answer: You would impose a tax that drives a wedge between prices paid by consumers p c
and prices paid by producers p s . Under the price floor, we calculated above that p c = 3.5
and p s = 3.278. If we therefore imposed a per unit tax t = (3.5 − 3.278) = 0.22, we would
achieve the same outcome.
(c) Suppose next that the government supplemented its price floor from (a) with a government
purchasing program that buys all surplus corn — and then sells it at the highest possible price
at which all surplus corn is bought. What is that price?
Answer: First, we have to calculate how much of a surplus the government would buy (on
top of the 9,111,111,111 bushels bought by consumers at the price floor). To do this, we can
calculate how much producers will produce at p = 3.5 by plugging this price into the supply
curve and solving for x. This gives us x s = 10,000,000,000. Thus, the government purchases
888,888,889 bushels. In order for a total of 10,000,000,000 bushels to be sold, the price of the
last bushel sold must be
p = 24 − 0.00000000225(10, 000, 000, 000) = 1.5.
(19.80)
(d) If you were to design a tax or subsidy policy that has the same impact on the marginal consumer and producer, what would it be?
Answer: We now have a consumer price for the marginal consumer of p c = 1.5 and a supplier price of p s = 3.5. To achieve this result, you could equally well impose a $2 per bushel
subsidy.
Distortionary Taxes and Subsidies
712
19.9 Policy Application: Rent Control: Is it a Tax or a Subsidy?: In exercise 18.11 we analyzed the impact
of rent control policies that impose a price ceiling in the housing rental market. The stated intent of such
policies is often to make housing more affordable. Before answering this question, you may wish to review
your answers to exercise 18.11.
A: Begin by illustrating the impact of the rent control price ceiling on the price received by landlords
and the eventual equilibrium price paid by renters.
(a) Why is it not an equilibrium for the price ceiling to be the rent actually paid by renters?
Answer: It is not an equilibrium because, at the rent controlled price, more people demand
apartments than are supplied — which implies some non-price rationing mechanism must
allocate the scarce apartments. This mechanism will raise the real cost of apartments to
consumers until demand is once again equal to supply.
(b) If you wanted to implement a tax or subsidy policy that achieves the same outcome as the rent
control policy, what policy would you propose?
Answer: This is illustrated in panel (a) of Graph 19.9 where the price ceiling results in a price
of p s received by suppliers and a price p c paid by consumers (once all costs of the nonprice rationing mechanism have been taken into account). We can achieve the same prices
for consumers and producers (as well as the same reduction in housing units provided) by
simply imposing a per-unit tax in the amount t = (p c − p s ).
Graph 19.9: Rent Control, Taxes and Subsidies
(c) Could you credibly argue that the alternative policy you proposed in (b) was designed to make
housing more affordable?
Answer: Since prices for consumers are raised from p ∗ to p c , it would not be possible to
argue this (since at least some of the tax will be passed to consumers).
(d) If you did actually want to make housing more affordable (rather than trying to replicate the
impact of rent control policies), would you choose a subsidy or a tax?
Answer: You would want to use a subsidy — which would result in p c < p ∗ < p s as well as
an increase in housing units.
(e) Illustrate your proposal from (d) — and show what would happen to the rental price received
by landlords and the rents paid by renters. What happens to the number of housing units
available for rent under your new policy?
Answer: This is illustrated in panel (b) of Graph 19.9 where a subsidy of s raises producer
prices to p s and lowers consumer prices to p c — while resulting in an increase in output.
The subsidy s can in fact be set so as to insure that housing will be exactly as affordable as
advocates of rent control wish when they set a price ceiling. (Under rent control, of course,
this would not be successful.)
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Distortionary Taxes and Subsidies
(f) True or False: Policies that make housing more affordable must invariably increase the equilibrium quantity of housing — and rent control policies fail because they reduce the equilibrium quantity of housing while subsidies succeed for the opposite reason.
Answer: This is true as illustrated above and in the Graph.
(g) True or False: Although rental subsidies succeed at the goal of making housing more affordable (while rent control policies fail to do so), we cannot in general say that deadweight loss
is greater or less under one policy rather than the other.
Answer: This is also true. In panel (a) of Graph 19.9, the deadweight loss from rent control is
at least (a +b) and may be as much as (a +b +c +d ) depending on the rationing mechanism
used. In panel (b) of the Graph, the deadweight loss from the subsidy is equal to area (e).
Depending on the relative elasticities of supply and demand, (e) may or may not be smaller
than (a + b). Of course the more the rationing mechanism under rent control causes deadweight loss to be larger than (a + b), the more likely it is that the subsidy will definitively be
more efficient. Still, the example illustrates that we may care more about the policy goal of
creating more affordable housing than the precise size of deadweight loss — and if that is
the case, then the subsidy is clearly the better policy as it actually achieves the policy goal
rather than doing the opposite.
B: Suppose, again as in exercise 18.11, that the aggregate monthly demand curve is p = 10000−0.01x
while the supply curve is p = 1000 + 0.002x. For simplicity, suppose again that there are no income
effects.
(a) Calculate the equilibrium number of apartments x ∗ and the equilibrium monthly rent p ∗ in
the absence of any price distortions.
Answer: Re-writing the demand and supply curves as demand and supply functions (by
writing them as functions of p), setting them equal to one another and solving, we get p ∗ =
2,500. Plugging this back into either the demand or supply function, we get x ∗ = 750,000.
(b) In exercise 18.11, you were asked to consider the impact of a $1,500 price ceiling. What housing tax or subsidy would result in the same economic impact?
Answer: Imposing a price ceiling of 1,500 will result in a reduction of x supplied. We can
solve for the quantity by substituting 1,500 for p into the supply curve and solving for x
to get x = 250,000. Thus, suppliers will supply 250,000 housing units at the price ceiling
of 1500. Consumers will then have to compete for the limited number of apartments —
exerting effort that adds to their cost of renting these units. In equilibrium, the effort cost
needs to be sufficient so that demand is equal to supply — which means the overall price
(including effort cost) must be
p c = 10000 − 0.01(250,000) = 7,500.
(19.81)
The rent control policy therefore results in a producer price of p s = 1,500 and a consumer
price of p c = 7,500 — a result we could equally well get by simply imposing a per unit tax of
$6,000.
(c) Suppose that you wanted to use tax/subsidy policies to actually reduce rents to $1,500 — the
stated goal of the rent control policy. What policy would you implement?
Answer: You would want to implement a subsidy program. In order to get the consumer
price to be 1500, we will need sufficient numbers of apartments so that every consumer
who wants to buy one at that price can get one. Thus, plugging 1,500 into the demand curve
and solving for x, we get that we need a total of 850,000 apartments. In order for producers
to supply that many apartments, they have to be able to charge
p s = 1000 + 0.002(850,000) = 2,700.
(19.82)
Thus, we need to provide a per unit subsidy of $1,200 in order to create sufficient incentives for the market to provide housing at a consumer price of $1,500 (and a seller’s price of
$2,700).
(d) Consider the policies you derived in (b) and (c). Under which policy is the deadweight loss
greater?
Distortionary Taxes and Subsidies
714
Answer: Graph 19.10 illustrates the supply and demand curves for this problem as well as
the various numbers we have calculated. The deadweight loss from the tax is the triangle
(a + b + c) to the left of the equilibrium while the deadweight loss from the subsidy is the
triangle (d + e) to the right of the equilibrium. (The deadweight loss under rent control is
at least the size of the deadweight loss from the tax but possibly more depending on the
rationing mechanism). It is easily seen in the picture that the deadweight loss from the tax
(and from rent control) is larger than the deadweight loss from the subsidy. (You can easily
verify that the tax deadweight loss triangle is $1,750,000,000 while the subsidy deadweight
loss triangle is $60,000,000.)
Graph 19.10: Rent Control Taxes versus Housing Subsidies
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Distortionary Taxes and Subsidies
19.10 Policy Application: Incidence of U.S. Social Security Taxes: In the U.S., the social security system
is funded by a payroll (wage) tax of 12.4% that is split equally between employer and employee; i.e. the
statutory incidence of the social security tax falls half on employers and half on employees.
A: In this exercise we consider how this split in statutory incidents impacts the labor market. Assume
throughout that labor supply is upward sloping.
(a) Illustrate the labor supply and demand graph and indicate the market wage w ∗ and employment level l ∗ in the absence of any taxes.
Answer: This is illustrated in Graqh 19.11 where the solid D and S curves are the labor demand and supply curves in the absence of taxes. These result in an equilibrium wage rate
w ∗ and employment level l ∗ .
Graph 19.11: Incidence of 2-part Social Security Tax
(b) Which curve shifts as a result of the statutory mandate that employers have to pay the government 6.2% of their wage bill? Which curve shifts because of the statutory mandate that
employees pay 6.2% of their wages in social security tax?
Answer: The statutory mandate on employers raises labor costs of employers — and thus
lowers demand for labor, shifting labor demand down. The statutory mandate on employees, on the other hand, causes workers to demand correspondingly more in wages — shifting labor supply up.
(c) Suppose the wage elasticity of labor demand and supply are equal in absolute value at the
pre-tax equilibrium. Can you illustrate how the market wage at the post-tax equilibrium —
when both parts of the social security tax are taken into account — might be unchanged from
the initial equilibrium wage w ∗ ?
Answer: This is also illustrated in Graph 19.11 where the supply curve shifts up to the dashed
S ′ and the demand curve shifts down to the dashed D ′ . The dashed labor demand and
supply curves intersect again at w ∗ but employment falls to l t .
(d) In your graph, illustrate what the imposition of the 2-part social security tax means for the
take home wage w w for workers. What does it mean for the real cost of labor w f that firms
incur?
Answer: Although the market wage remains unchanged at w ∗ , workers now have to pay
their statutory part of the social security tax, which effectively lowers their take-home pay
below w ∗ . Similarly, employers still have to pay their statutory share of the social security
Distortionary Taxes and Subsidies
716
tax — thus incurring real labor cost above w ∗ . To be more precise, the additional statutory
obligation incurred by workers is equal to the upward shift in the labor supply curve (from
S to S ′ ) — which is equal to the vertical distance between B and C in the graph. Thus, the
take-home wage for workers falls to w w . The additional statutory obligation for firms, on
the other hand, is equal in size to the downward shift of demand (from D to D ′ ) — which is
equal to the vertical distance from B to E. Thus, the real labor cost for firms increases to w f
in the graph.
(e) How would the equilibrium wage in the market change if the government imposed the entire
12.4% tax on workers (and let employers statutorily off the hook)? How would it change if the
government instead imposed the entire tax on employers?
Answer: In the case where the entire statutory incidence is placed on employees, the supply
curve would shift up by the entire distance of the tax (and not just half of it) — which would
cause the new supply curve to intersect the unchanged labor demand curve D at point E.
Thus, the equilibrium wage observed in the market would rise to w f in the graph. If the
statutory incidence of the tax were placed entirely on employers, on the other hand, the demand curve would shift down by the entire distance of the tax (and not just half) — causing
it to intersect the unchanged supply curve S at point C . The equilibrium wage observed in
the market would therefore be w w .
(f) What happens to the take-home wage for workers and the real labor cost of firms as a result
of the two statutory tax reforms raised in part (e)?
Answer: These would be unchanged. In the case where the entire statutory incidence of the
tax falls on workers and we end at the new equilibrium E in Graph 19.11, workers receive
w f from firms but still have to pay the entire social security tax — causing their take-home
wage to fall to w w . (Firms would be paying w f to employees — implying their labor costs
would be w f .) In the case where the statutory incidence is placed on employers, we end up
at equilibrium C in the graph — with workers being paid w w (which is then their take-home
pay) and firms still having to pay the entire tax to the government. Thus, the real labor cost
for firms is again w f .
(g) Does any of this analysis depend on whether there are wealth effects in the labor market?
Answer: No, the entire analysis is based on actual market demand and supply curves because we are predicting actual wages and employment levels. The only way wealth effects
would matter is if we wanted to use these curves to assign worker surplus in our graphs —
which we did not do.
B: Suppose, as in exercise 19.6, that labor demand and supply in the absence of taxes are given by
l d = (A − w)/α and l s = (w − B )/β.
(a) Determine the equilibrium employment level l ∗ and the equilibrium wage w ∗ .
Answer: Setting l d equal to l s and solving for w, we get the equilibrium wage rate
βA + αB
.
α+β
Substituting this into either the demand or supply equation, we then get
w∗ =
(19.83)
A −B
(19.84)
.
α+β
(b) Now suppose the government imposes a per-unit tax t on workers and a second per-unit tax t
on employers. Derive the new labor demand and supply curves that incorporate these (as you
would when you shift demand and supply curves in response to statutory tax laws).
Answer: The demand and supply curves are the demand and supply functions solved for p.
This implies that A is the vertical intercept of the labor demand curve and B is the vertical intercept of the labor supply curve. When these statutory tax rates are imposed, labor
demand shifts down by t and labor supply shifts down by t . Put differently, the vertical intercept A of the demand curve becomes (A − t ) while the vertical intercept B of the labor
supply curve becomes (B + t ). The labor demand and supply functions thus become
l∗ =
ld =
w − (B + t )
(A − t ) − w
and l s =
.
α+β
α+β
(19.85)
717
Distortionary Taxes and Subsidies
(c) Determine the new equilibrium wage and employment level. Under what condition is the
new observed equilibrium wage unchanged as a result of the two-part wage tax? Is there any
way that employment will not fall?
Answer: Setting the equations in (19.85) equal to one another and solving for w, we get a
new equilibrium wage rate
w t∗ =
βA + αB + (α − β)t
.
α+β
(19.86)
Plugging this into either of the equations, we get
l t∗ =
A − B − 2t
.
α+β
(19.87)
The wage rate w t∗ is equal to the pre-tax equilibrium wage w ∗ if and only if α = β. The
employment level unambiguously falls for any t > 0.
(d) Determine the take-home wage w w for workers and the real labor cost w f for firms.
Answer: Under the equal statutory incidence, both workers and firms have to still pay t per
wage hour after the equilibrium wage w t∗ is transacted. Thus
w w = w t∗ − t =
βA + αB − 2βt
βA + αB + (α − β)t
−t =
α+β
α+β
(19.88)
w f = w t∗ + t =
βA + αB + 2αt
βA + αB + (α − β)t
+t =
.
α+β
α+β
(19.89)
and
(e) Suppose you did not know the statutory incidence of the wage tax but simply knew the total
tax was equal to 2t . How would you calculate the economic incidence — i.e. how would you
calculate w w and w f ?
Answer: In this case, we would simply use the fact that, in the post-tax equilibrium, it has to
be the case that workers take home 2t less than firms pay — i.e. w w = w f −2t . We can then
write labor demand and supply functions as
ld =
A−wf
α
and l s =
w f − 2t − B
ww − B
=
.
β
β
(19.90)
Setting these equal to each other and solving for w f , we get
wf =
βA + αB + 2αt
α+β
(19.91)
and plugging this into w w = w f − 2t , we get
ww =
βA + αB − 2βt
.
α+β
(19.92)
(f) Compare your answers to (e) to your answers to (d). Can you conclude from this whether
statutory incidence matters?
Answer: The answers are identical. Statutory incidence does not matter for how workers
and employers are affected — without knowing statutory incidence we arrive at the same
answer as when we do know it and use that information to shift demand and supply curves.
It is true that the observed equilibrium wage depends on how the tax law is written — but
neither the take-home wage nor the firms’ labor costs depend on this once we take into
account the tax payments that have to be made once the equilibrium wage has been transacted.
Distortionary Taxes and Subsidies
718
19.11 Policy Application: Mortgage Interest Deductibility, Land Values and the Equilibrium Rate of Return on Capital: In the text, we suggested that the property tax can be thought of in part as a tax on land
and in part as a tax on capital invested in housing. In the U.S., property taxes are typically levied by local governments — while the major piece of federal housing policy is contained in the federal income tax
code which allows individuals to deduct (from income) the interest they pay on home mortgages prior to
calculating the amount of taxes owed.
A: Whereas we can think of the property tax as a tax on both land and housing structures, we can
think of the homeownership subsidy in the federal tax code as a subsidy on land and housing structures.
(a) If your marginal federal income tax rate is 25% and you are financing 100% of your home
value, how much of your housing consumption is being subsidized through the tax code?
What if you are only financing 50% of the value of your home?
Answer: In the first scenario, about 25% of your housing consumption is subsidized by the
government through the tax code (because almost all of your mortgage payments are tax
deductible since most of your payment is made up of interest that is deductible). If you are
only financing 50%, only about 12.5% of your housing consumption is subsidized.
(b) Suppose homeowners are similar to one another in terms of their marginal tax rate and how
much of their home they are financing, and suppose that this implies a subsidy of s for every
dollar of housing/land consumption. How would you predict the value of suburban residential land (assumed to be in fixed supply) is different as a result of this than it would have been
in the absence of this policy?
Answer: Graph 19.12 illustrates the market for residential land initially denominated in
units such that the original equilibrium price (at A) is $1. The effective subsidy s per dollar
then raises the rental price of residential land to (1 + s) — with the entire incidence of the
subsidy going to land owners — i.e. land owners now collect (1 + s) per unit of land that
they originally collected just $1 on. We would therefore predict that residential land value
increases by the present discounted value of all future streams of s for each unit of land.
Graph 19.12: Tax Code Subsidy for Residential Land
(c) When s was first introduced, who benefitted from the implicit land subsidy: current homeowners or future homeowners?
Answer: Current homeowners (i.e. those who owned homes when the subsidy was introduced) benefitted — because all increases in rents from land (due to the subsidy) are immediately incorporated into the land price. If you owned land, your asset therefore became
more valuable. If you did not own land when the subsidy was introduced, then you would
not benefit from the subsidy because you will pay for the benefit of the subsidy streams
719
Distortionary Taxes and Subsidies
when you buy land at the value that already incorporates all those future streams of subsidies.
(d) Now consider s as a subsidy on housing capital. Do you think houses are larger or smaller as
a result of the federal income tax code?
Answer: This subsidy on housing capital reduces the price of building on land — and therefore will cause an increase in housing consumption (so long as housing is not a Giffen good).
We would therefore tend to think that houses are larger as a result of the federal income tax
code.
(e) Suppose that the overall amount of capital in the economy is fixed and that capital is mobile
across sectors. Thus, any given unit of capital can be invested in housing or alternatively
in some other non-housing sector where it earns some rate of return. If the overall amount of
capital in the economy is fixed, what happens to the fraction of capital invested in the housing
sector?
Answer: If houses are larger as a result of the tax subsidy (as we just concluded), this means
that more capital is invested in housing than would otherwise be the case. Thus, the fraction
of capital invested in the housing sector increases.
(f) What would you predict will happen to the rate of return on capital in the non-housing sector? Explain.
Answer: As capital shifts from the non-housing sector to the housing sector, the marginal
product of capital must increase in the non-housing sector — which implies the rate of
return on capital increases in the non-housing sector. Thus, the benefit of the subsidy on
housing capital (as opposed to the subsidy on land) accrues to all forms of capital, not just
housing capital.
(g) True or False: Even though only housing capital is statutorily subsidized, the economic incidence of this subsidy falls equally on all forms of capital (so long as capital is mobile between
sectors).
Answer: This is true as just explained in the previous part.
B: Suppose we model owners of capital as a “representative investor” who chooses to allocate K units
of capital between the housing sector and other sectors of the economy. With k1 representing capital
invested in housing and k2 representing capital invested in other sectors, suppose f 1 (k1 ) = αk10.5
and f 2 (k2 ) = βk20.5 are the production functions of the two sectors.
(a) In the absence of any policy distortions, calculate the fraction of total capital (K ) that is invested in the housing sector.
Answer: Our representative investor then wants to maximize her total return by optimally
choosing the allocation of her capital K across the two sectors. Put differently, she wants to
solve the maximization problem
max f 1 (k1 ) + f 2 (k2 ) subject to k1 + k2 = K .
k 1 ,k 2
(19.93)
The solution to this problem is
k1∗ =
α2 K
α2 + β2
and k2∗ =
β2 K
α2 + β2
.
(19.94)
(Note that at this solution, the marginal product of capital is the same in the two sectors).
The fraction of total capital invested in housing is then (α2 /(α2 + β2 ).
(b) What changes as a result of the federal income tax code’s implicit housing subsidy s.
Answer: The problem then becomes
max (1 + s) f 1 (k1 ) + f 2 (k2 ) subject to k1 + k2 = K .
k 1 ,k 2
(19.95)
The solution to this problem is
k1∗ =
(1 + s)2 α2 K
(1 + s)2 α2 + β2
β2 K
and k2∗ =
.
(1 + s)2 α2 + β2
(19.96)
Distortionary Taxes and Subsidies
720
Thus, the fraction of capital invested in the non-housing sector falls while the fraction invested in the housing sector increases.
(c) What happens to the marginal product of capital in the non-housing sector?
Answer: The marginal product of capital in the non-housing sector is given by
MP K 2 =
∂ f 2 (k2 )
= 0.5βk −0.5 .
∂k2
(19.97)
Plugging k2 from equation (19.96) into this, we get
MP K 2 = 0.5β
Ã
β2 K
(1 + s)2 α2 + β2
!−0.5
= 0.5
Ã
!
(1 + s)2 α2 + β2
.
K
(19.98)
Thus, as s increases, the marginal product of non-housing capital also increases.
(d) What happens to the equilibrium rate of return on capital?
Answer: In equilibrium, the marginal product of capital in the non-housing sector increases
(as just demonstrated) until it is equal to the subsidy-inclusive marginal product of capital
in the housing sector — thus, the equilibrium marginal return on capital increases across
all sectors.
(e) True or False: The general equilibrium subsidy incidence of the implicit subsidy of housing
capital falls equally on all forms of capital.
Answer: This is true — capital moves between the sectors until the rate of return is equalized.
721
Distortionary Taxes and Subsidies
19.12 Policy Application: The Split-Rate Property Tax: As we have mentioned several times, the usual
property tax is really two taxes: one levied on land value (or on land rents) and the other levied on the value
of the “improvements” of land — or the rents from capital investments. The typical property tax simply sets
the same tax rate for each part, but in an increasing number of places, governments are reforming property
taxes to levy a higher rate on land than on improvements. Such a tax is called a split-rate property tax.
A: Suppose you are in a locality that currently taxes rental income from capital at the same rate as
rental income from land. Assume throughout that the amount of land in the community is fixed.
(a) Which portion of your local tax system is distortionary and which is non-distortionary?
Answer: The portion that imposes taxes on capital is distortionary while the portion that
imposes taxes on land is non-distortionary. This is because the overall amount of land in the
community is fixed — which implies that any tax on land rents will simply lower the wealth
of land owners and act as a lump sum tax on them. But a tax on capital distorts investment
decisions since capital can be invested here or elsewhere — and the tax is imposed only
locally.
(b) Next, suppose that your community lowers the tax on capital income and raises it on land
rents — and suppose that overall tax revenues are unchanged as a result of this reform. Do
you think the tax reform enhances efficiency?
Answer: Yes, this must enhance efficiency (unless capital and land are perfect complements.) Again, this is because we are reducing a distortionary tax and replacing it with a
non-distortionary tax.
(c) Your community has a fixed amount of land — but capital can move in and out of your community and therefore changes depending on economic conditions. Do you think the land in
your community will be more or less intensively utilized as a result of the tax reform — i.e. do
you think more or less capital will be invested on it?
Answer: It will be more intensively utilized because the cost of using capital has decreased
with a decrease in the local tax on capital. Put differently, the necessary return on capital
in order for investments to occur is now lower than it was before — because capital owners
do not have to pay as much tax on their capital income. Thus, they will be willing to invest
capital farther down the marginal product of capital curve.
(d) What do you think happens to the marginal product of land in your community under this
tax reform? What must therefore happen to the rental value of land (before land rent taxes
are paid)?
Answer: Since more capital is used on each plot of land, it must be that the marginal product
of land increases. As a result, the rental value of land must increase.
(e) Suppose half of your community has land that is relatively substitutable with capital in production — and the other half of your community has land that is relatively complementary to
capital in production. Might it be the case that land values go up in part of your community
and go down in another part of your community as a result of the tax reform? If so, which
part experiences the increase in land values despite an increase in the tax on land rents?
Answer: Yes, this might be the case. The tax reform is more valuable the more substitutable
capital and labor are — because reducing the distortion on capital investments will cause a
sharper increase in capital on land that is relatively substitutable for capital. Thus, if land
values increase in some parts of town and decrease in others, we would expect the increase
to happen where land is more substitutable for capital.
(f) Will overall output in your community increase or decrease as a result of the tax reform? Under what extreme assumption about the degree of substitutability of land and capital in production would local production remain unchanged?
Answer: The amount of land is fixed but the amount of capital increases with the tax reform.
Thus, with one input remaining constant and the other increasing, output must increase.
The only case in which this would not happen is if capital and land are perfect complements
in production — in which case there would be no increase in capital as the tax on capital
falls.
(g) True or False: The more substitutable land and capital are in production, the more likely it is
that the tax reform toward a split-rate property tax (that taxes land more heavily) will result
in a Pareto improvement.
Distortionary Taxes and Subsidies
722
Answer: This is true. Even though the tax reform is efficient, it may not result in a Pareto
improvement (i.e. in everyone becoming better off) if land values drop (and thus landowners are hurt) due to the increase tax on land rents. This is more likely to happen the more
complementary land and capital are — because with such strong complementarities, the
increase in capital intensity on the land will not raise land rents sufficiently to cause land
values to increase once we take into account that taxes on land rents have increased. But if
capital and land are very substitutable, then the inflow of capital from a decrease in the tax
on capital can be sufficient to cause a sufficiently large increase in land rents to cause land
values to also increase (despite the fact that land rents are being taxed more heavily).
B: Suppose we normalize units of land so that the entire land area of a particular locality equals one
unit. Economic activity is captured by the constant elasticity of substitution production function
y = f (k,L) = (0.5L −1 + 0.5k −1 )−1 . The government collects revenues through a property tax that
taxes land rents at a rate t L and the rental value of capital at a rate t k — resulting in total tax
revenue of T R = t L R + t k r K where R is the rental value of the 1 unit of land in the locality, r is the
interest rate in the local economy and K is the total capital employed in the locality. (Note that we
have defined capital units such that the interest rate is equal to the rental rate of capital).
(a) Suppose that this locality is sufficiently small so that nothing it does can affect the global
economy’s rental rate r — i.e. the supply of capital is perfectly elastic. If the locality taxes the
rental value of capital at rate t k , at what local interest rate r would investors be willing to
invest here?
Answer: The lowest local interest rate r an investor will accept is one that makes his after-tax
return (1 − t k )r equal to the return r he can get elsewhere; i.e. (1 − t k )r = r or
r=
r
.
(1 − t k )
(19.99)
(b) Suppose that land is utilized optimally given the local tax environment — which implies that
the marginal product of capital must equal r . Define the equation that you would have to
solve in order to calculate the level of capital invested in this locality.
Answer: For a fixed level of land equal to L = 1, we would have to solve MP k = r . With L = 1,
the marginal product of capital is the derivative of f (k) = (0.5 + 0.5k −1 )−1 with respect to k
—
¶2
³
´−2 0.5 µ
k
2
.
=
MP k = 0.5k −2 0.5 + 0.5k −1
= 2
0.5(k + 1)
k
(k + 1)2
(19.100)
Setting this equal to our expression for r , we then have to solve the equation
2
(k + 1)2
=
r
.
(1 − t k )
(19.101)
(c) Suppose r = 0.06. Solve for the level of capital K invested on the one unit of land of this
locality (as a function of t k ).
Answer: Plugging r = 0.06 into our equation and solving for k, we get
K=
µ
¶
100(1 − t k ) 1/2
− 1 ≈ 5.7735(1 − t k )1/2 − 1.
3
(19.102)
(d) Can you determine the rental value of land? (Hint: Derive the marginal product of land and
evaluate it at the level of capital you calculated in the previous part and the 1 unit of land
that is available.)
Answer: First, we derive the marginal product of land
MP L =
which becomes
³
´−2
2k 2
∂ f (k,L)
= 0.5L −2 0.5L −1 + 0.5k −1
=
∂L
(k + L)2
MP L =
2k 2
(k + 1)2
(19.103)
(19.104)
723
Distortionary Taxes and Subsidies
(tL , tk )
K
R
P
y
TR
(0, 0.5)
3.0825
1.1402
19.003
1.5101
0.1850
Split-Rate Property Taxes
(0.05, 0.3637) (0.1, 0.1748)
3.605
4.2447
1.2258
1.3100
19.408
19.650
1.5657
1.6187
0.1850
0.1850
(0.1353, 0)
4.7735
1.3672
19.703
1.6536
0.1850
Table 19.1: Revenue Neutral Property Tax Reforms
when L = 1. Substituting K from equation (19.102), we get the rental value of land
³
´2
³
´2
0.06 5.7735(1 − t k )1/2 − 1
2 5.7735(1 − t k )1/2 − 1
.
R= ¡
¢2 =
(1 − t k )
5.7735(1 − t k )1/2 − 1 + 1
(19.105)
(e) Now consider the case where the local tax system is (t L , t k ) = (0,0.5). Derive the total capital
K invested in the locality, the land rental value R, the value of land P (assuming that future
income is discounted at the interest rate r = 0.06), the production level y and the tax revenue
T R. (You may find it convenient to set up a simple spreadsheet to do the calculations for you).
Answer: Plugging t k = 0.5 into equations (19.102) and (19.105), we get K ≈ 3.0825 and R ≈
1.1402. The value of land is then simply
1.1402
≈ 19,
(19.106)
0.06
and output is given by plugging in L = 1 and k = 3.0825 into the production function to get
P=
Finally, tax revenue is
³
´−1
y = f (1,3.0825) = 0.5 + 0.5(3.0835)−1
≈ 1.51.
T R = t k r K = 0.5
µ
¶
0.06
3.0835 ≈ 0.1850.
(1 − 0.5)
(19.107)
(19.108)
(f) Repeat this for the tax system (t L , t k ) = (0.05,0.3637), the tax system (t L , t k ) = (0.1,0.1748)
and the tax system (t L , t k ) = (0.1353,0). Present your results for K , R, P , y and T R in a table
(and keep in mind that r changes with t k even though r remains at 0.06.) (Hint: All three
systems should give the same tax revenue.)
Answer: These are presented in Table 19.1.
(g) Use your table to discuss how the shift from a tax solely on capital (i.e. structures) toward
a revenue-neutral tax system that increasingly relies on taxing land rents impacts the local
economy. Which of the rows in your table could look qualitatively different under different
elasticity of substitution assumptions?
Answer: As we move across in the table, we see a decrease in the distortionary tax on capital
and an increase in the non-distortionary tax on land rents. As the tax on capital falls (going
from left to right), we see the amount of capital invested increase. Because of this increased
capital utilization of land, the marginal product of land increases — implying the the rental
value R increases. In fact, for the simulations here, this increase in the rental value of the
land is sufficient to cause P — the market value of the land — to increase even though land
rents are taxed as we move from left to right. Put differently, the decreased distortion from
the lower capital tax creates sufficient benefit such that land becomes more valuable even
though it is itself taxed more heavily. Finally, we see output increase as the tax system shifts
from capital to land taxes and the distortion in the tax system is slowly removed. All of these
rows should move in the same direction regardless of the elasticity of substitution of capital
Distortionary Taxes and Subsidies
724
for land — except for the P row. As land and capital become less substitutable, the benefit
to land owners from lower levels of distortionary taxation on capital may not be sufficient
to offset the increase in the tax on land rents.